Analytic Bootstrap for Boundary CFT

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator $⟨\varphi \varphi ⟩$ in ${\varphi }^4$ theory at the Wilson-Fisher fixed point is computed to order ${ϵ}^2$ in the $ϵ$ expansion.